Optimal. Leaf size=21 \[ -\frac {3 \log (x)}{2}+4 \log (1+x)-\frac {5}{2} \log (2+x) \]
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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1608, 814}
\begin {gather*} -\frac {3 \log (x)}{2}+4 \log (x+1)-\frac {5}{2} \log (x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 814
Rule 1608
Rubi steps
\begin {align*} \int \frac {-3+x}{2 x+3 x^2+x^3} \, dx &=\int \frac {-3+x}{x \left (2+3 x+x^2\right )} \, dx\\ &=\int \left (-\frac {3}{2 x}+\frac {4}{1+x}-\frac {5}{2 (2+x)}\right ) \, dx\\ &=-\frac {3 \log (x)}{2}+4 \log (1+x)-\frac {5}{2} \log (2+x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} -\frac {3 \log (x)}{2}+4 \log (1+x)-\frac {5}{2} \log (2+x) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.70, size = 17, normalized size = 0.81 \begin {gather*} \frac {-5 \text {Log}\left [2+x\right ]}{2}-\frac {3 \text {Log}\left [x\right ]}{2}+4 \text {Log}\left [1+x\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 18, normalized size = 0.86
method | result | size |
default | \(-\frac {3 \ln \left (x \right )}{2}+4 \ln \left (1+x \right )-\frac {5 \ln \left (2+x \right )}{2}\) | \(18\) |
norman | \(-\frac {3 \ln \left (x \right )}{2}+4 \ln \left (1+x \right )-\frac {5 \ln \left (2+x \right )}{2}\) | \(18\) |
risch | \(-\frac {3 \ln \left (x \right )}{2}+4 \ln \left (1+x \right )-\frac {5 \ln \left (2+x \right )}{2}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 17, normalized size = 0.81 \begin {gather*} -\frac {5}{2} \, \log \left (x + 2\right ) + 4 \, \log \left (x + 1\right ) - \frac {3}{2} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 17, normalized size = 0.81 \begin {gather*} -\frac {5}{2} \, \log \left (x + 2\right ) + 4 \, \log \left (x + 1\right ) - \frac {3}{2} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 20, normalized size = 0.95 \begin {gather*} - \frac {3 \log {\left (x \right )}}{2} + 4 \log {\left (x + 1 \right )} - \frac {5 \log {\left (x + 2 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 24, normalized size = 1.14 \begin {gather*} -\frac {3}{2} \ln \left |x\right |+4 \ln \left |x+1\right |-\frac {5}{2} \ln \left |x+2\right | \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 17, normalized size = 0.81 \begin {gather*} 4\,\ln \left (x+1\right )-\frac {5\,\ln \left (x+2\right )}{2}-\frac {3\,\ln \left (x\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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